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class="layout" id="content-inner"><div id="post"><article class="post-content" id="article-container"><h1 id="10-4-变态跳台阶"><a href="#10-4-变态跳台阶" class="headerlink" title="10.4 变态跳台阶"></a>10.4 变态跳台阶</h1><h2 id="题目链接"><a href="#题目链接" class="headerlink" title="题目链接"></a>题目链接</h2><p><a target="_blank" rel="noopener" href="https://www.nowcoder.com/practice/22243d016f6b47f2a6928b4313c85387?tpId=13&tqId=11162&tPage=1&rp=1&ru=/ta/coding-interviews&qru=/ta/coding-interviews/question-ranking&from=cyc_github">牛客网</a></p>
<h2 id="题目描述"><a href="#题目描述" class="headerlink" title="题目描述"></a>题目描述</h2><p>一只青蛙一次可以跳上 1 级台阶，也可以跳上 2 级… 它也可以跳上 n 级。求该青蛙跳上一个 n 级的台阶总共有多少种跳法。</p>
<div align="center"> <img src= "" data-lazy-src="https://cs-notes-1256109796.cos.ap-guangzhou.myqcloud.com/cd411a94-3786-4c94-9e08-f28320e010d5.png" width="380px"> </div><br>

<h2 id="解题思路"><a href="#解题思路" class="headerlink" title="解题思路"></a>解题思路</h2><h3 id="动态规划"><a href="#动态规划" class="headerlink" title="动态规划"></a>动态规划</h3><figure class="highlight java"><table><tr><td class="code"><pre><span class="line"><span class="function"><span class="keyword">public</span> <span class="keyword">int</span> <span class="title">jumpFloorII</span><span class="params">(<span class="keyword">int</span> target)</span> </span>&#123;</span><br><span class="line">    <span class="keyword">int</span>[] dp = <span class="keyword">new</span> <span class="keyword">int</span>[target];</span><br><span class="line">    Arrays.fill(dp, <span class="number">1</span>);</span><br><span class="line">    <span class="keyword">for</span> (<span class="keyword">int</span> i = <span class="number">1</span>; i &lt; target; i++)</span><br><span class="line">        <span class="keyword">for</span> (<span class="keyword">int</span> j = <span class="number">0</span>; j &lt; i; j++)</span><br><span class="line">            dp[i] += dp[j];</span><br><span class="line">    <span class="keyword">return</span> dp[target - <span class="number">1</span>];</span><br><span class="line">&#125;</span><br></pre></td></tr></table></figure>

<h3 id="数学推导"><a href="#数学推导" class="headerlink" title="数学推导"></a>数学推导</h3><p>跳上 n-1 级台阶，可以从 n-2 级跳 1 级上去，也可以从 n-3 级跳 2 级上去…，那么</p>
<figure class="highlight plaintext"><table><tr><td class="code"><pre><span class="line">f(n-1) = f(n-2) + f(n-3) + ... + f(0)</span><br></pre></td></tr></table></figure>

<p>同样，跳上 n 级台阶，可以从 n-1 级跳 1 级上去，也可以从 n-2 级跳 2 级上去… ，那么</p>
<figure class="highlight plaintext"><table><tr><td class="code"><pre><span class="line">f(n) = f(n-1) + f(n-2) + ... + f(0)</span><br></pre></td></tr></table></figure>

<p>综上可得</p>
<figure class="highlight plaintext"><table><tr><td class="code"><pre><span class="line">f(n) - f(n-1) = f(n-1)</span><br></pre></td></tr></table></figure>

<p>即</p>
<figure class="highlight plaintext"><table><tr><td class="code"><pre><span class="line">f(n) = 2*f(n-1)</span><br></pre></td></tr></table></figure>

<p>所以 f(n) 是一个等比数列</p>
<figure class="highlight plaintext"><table><tr><td class="code"><pre><span class="line">public int JumpFloorII(int target) &#123;</span><br><span class="line">    return (int) Math.pow(2, target - 1);</span><br><span class="line">&#125;</span><br></pre></td></tr></table></figure>
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class="relatedPosts-list"><div><a href="/hexo3/2021/12/06/notes/10.1%20%E6%96%90%E6%B3%A2%E9%82%A3%E5%A5%91%E6%95%B0%E5%88%97/" title="10.1 斐波那契数列"><img class="cover" src= "" data-lazy-src="/hexo3/img/16.jpg" alt="cover"><div class="content is-center"><div class="date"><i class="far fa-calendar-alt fa-fw"></i> 2021-12-06</div><div class="title">10.1 斐波那契数列</div></div></a></div><div><a href="/hexo3/2021/12/06/notes/10.2%20%E7%9F%A9%E5%BD%A2%E8%A6%86%E7%9B%96/" title="10.2 矩形覆盖"><img class="cover" src= "" data-lazy-src="/hexo3/img/19.jpg" alt="cover"><div class="content is-center"><div class="date"><i class="far fa-calendar-alt fa-fw"></i> 2021-12-06</div><div class="title">10.2 矩形覆盖</div></div></a></div><div><a href="/hexo3/2021/12/06/notes/10.3%20%E8%B7%B3%E5%8F%B0%E9%98%B6/" title="10.3 跳台阶"><img class="cover" src= "" data-lazy-src="/hexo3/img/17.jpg" alt="cover"><div class="content is-center"><div class="date"><i class="far fa-calendar-alt fa-fw"></i> 2021-12-06</div><div 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src= "" data-lazy-src="/hexo3/img/2.jpg" alt="cover"><div class="content is-center"><div class="date"><i class="far fa-calendar-alt fa-fw"></i> 2021-12-06</div><div class="title">48. 最长不含重复字符的子字符串</div></div></a></div></div></div></div><div class="aside-content" id="aside-content"><div class="sticky_layout"><div class="card-widget" id="card-toc"><div class="item-headline"><i class="fas fa-stream"></i><span>目录</span></div><div class="toc-content"><ol class="toc"><li class="toc-item toc-level-1"><a class="toc-link" href="#10-4-%E5%8F%98%E6%80%81%E8%B7%B3%E5%8F%B0%E9%98%B6"><span class="toc-number">1.</span> <span class="toc-text">10.4 变态跳台阶</span></a><ol class="toc-child"><li class="toc-item toc-level-2"><a class="toc-link" href="#%E9%A2%98%E7%9B%AE%E9%93%BE%E6%8E%A5"><span class="toc-number">1.1.</span> <span class="toc-text">题目链接</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#%E9%A2%98%E7%9B%AE%E6%8F%8F%E8%BF%B0"><span class="toc-number">1.2.</span> <span class="toc-text">题目描述</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#%E8%A7%A3%E9%A2%98%E6%80%9D%E8%B7%AF"><span class="toc-number">1.3.</span> <span class="toc-text">解题思路</span></a><ol class="toc-child"><li class="toc-item toc-level-3"><a class="toc-link" href="#%E5%8A%A8%E6%80%81%E8%A7%84%E5%88%92"><span class="toc-number">1.3.1.</span> <span class="toc-text">动态规划</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#%E6%95%B0%E5%AD%A6%E6%8E%A8%E5%AF%BC"><span class="toc-number">1.3.2.</span> <span class="toc-text">数学推导</span></a></li></ol></li></ol></li></ol></div></div></div></div></main><footer id="footer" style="background-image: url('/hexo3/img/1.jpg')"><div id="footer-wrap"><div class="copyright">&copy;2020 - 2022 By Zhang Shuo</div><div class="framework-info"><span>框架 </span><a target="_blank" rel="noopener" href="https://hexo.io">Hexo</a><span class="footer-separator">|</span><span>主题 </span><a target="_blank" rel="noopener" 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